幂零群的平均Dehn函数
Averaged Dehn Functions for Nilpotent Groups
摘要：
格罗莫夫提出了一个平均版本的德恩函数，并声称在许多情况下，它应该是次渐近的德恩函数。利用幂零群中随机游动的结果，我们对大多数幂零群证实了这一结论。特别地，如果幂零群满足$α>2$的等周不等式$δ(l)<Cl^α$，则它满足平均等周不等式$δ^{\text{avg}}(l)<C'l^{α/2}$。在非交换自由幂零群群的情况下，我们给出的界是渐近尖锐的。
英文摘要：
Gromov proposed an averaged version of the Dehn function and claimed that in many cases it should be subasymptotic to the Dehn function. Using results on random walks in nilpotent groups, we confirm this claim for most nilpotent groups. In particular, if a nilpotent group satisfies the isoperimetric inequality $δ(l)<Cl^α$ for $α>2$ then it satisfies the averaged isoperimetric inequality $δ^{\text{avg}}(l)<C'l^{α/2}$. In the case of non-abelian free nilpotent groups, the bounds we give are asymptotically sharp.
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